# Conjugate elements

From Groupprops

*This article describes an equivalence relation on the set of elements of a group*

## Contents

## Definition

### Symbol-free definition

Given a group, two (possibly equal) elements of the group are termed **conjugate elements** if the following equivalent conditions are satisfied:

- There is an inner automorphism of the group mapping one element to the other
- There are two elements of the group whose products, in the two possible orders, give these two elements

### Definition with symbols

Given a group and elements , is termed conjugate to if the following equivalent conditions are satisfied:

- There exists such that , in other words, the inner automorphism of conjugation by , sends to
- There exist such that

The equivalence classes under the equivalence relation of being conjugate are termed the conjugacy classes.

### Equivalence of definitions

`For full proof, refer: Equivalence of definitions of conjugate elements`